Formulating tightest bounds on causal effects in studies with unmeasured confounders.

This paper considers the problem of evaluating the causal effect of an exposure on an outcome in observational studies with both measured and unmeasured confounders between the exposure and the outcome. Under such a situation, MacLehose et al. (Epidemiology 2005; 16:548-555) applied linear programming optimization software to find the minimum and maximum possible values of the causal effect for specific numerical data. In this paper, we apply the symbolic Balke-Pearl linear programming method (Probabilistic counterfactuals: semantics, computation, and applications. Ph.D. Thesis, UCLA Cognitive Systems Laboratory, 1995; J. Amer. Statist. Assoc. 1997; 92:1172-1176) to derive the simple closed-form expressions for the lower and upper bounds on causal effects under various assumptions of monotonicity. These universal bounds enable epidemiologists and medical researchers to assess causal effects from observed data with minimum computational effort, and they further shed light on the accuracy of the assessment.